Practical Geometry - 8th math

Class 8 math Practical Geometry Construction of quadrilateral Solution of NCERT Exercise 4.4 Question (1) Construct the following quadrilaterals.

(i) Quadrilateral DEAR

DE = 4 cm

EA = 5 cm

AR = 4.5 cm

∠E = 60^o

∠A = 90^o

Solution

First of all draw a rough sketch of the given quadrilateral to estimate the idea of construction.

Step (i) Draw a horizontal line to draw DE and cut 4 cm from it using a compass and a measuring scale.

Step (ii) At point E draw an angle of 60^o. And draw an arc of radius 5 cm from it.

Step (iii) Draw and angle of 90^o at point A. Now draw and arc of radius 4.5 cm to draw the side AR = 4.5 cm.

Step (iv) Join the points R and D to draw the side of given quadrilateral RD and complete the construction.

This is the construction of given quadrilateral.

Class 8 math Practical Geometry Construction of quadrilateral Solution of NCERT Exercise 4.4 Question (1) (ii) Quadrilateral TRUE

TR = 3.5 cm

RU = 3 cm

UE = 4 cm

∠R = 75^o

∠U = 120^o

Solution

First of all a rough sketch of given quadrilateral is made to estimate its construction.

Step (i) Draw a horizontal line and mark 3.5 cm from it by making two arcs.

Step (ii) Draw and angle of 75^o at point R. And draw an arc of radius 3 cm from line of this angle to make the side RU

Step (iii) Draw an angle of 120^o at point U. And draw and arc of radius 4 cm on the line of this angle to draw the side EU.

Step (iv) Now, join points E and T and complete the construction of given quadrilateral.

This is the construction of the given quadrilateral.

Think Discuss and Write

Think Discuss and Write (1) In the above example, we first drew BC. Instead, what could have been be the other starting points?

Solution

Question in the example refer in the question is Example (4) Construct a quadrilateral ABCD, where AB = 4 cm, BC = 5 cm, CD = 6.5 cm and ∠B = 105^o and ∠C = 80^o

In the solved example given quadrilateral is constructed taking BC = cm as starting point.

This quadrilateral can be constructed using AB = 4 cm or DC = 6.5 cm as starting point.

Construction of given quadrilateral taking AB as starting point

Here is the construction.

First of all a rough sketch of the given quadrilateral with given statistics is drawn to estimate the construction.

Construction starting with side AB = 4 cm

Step (i) Draw a horizontal line of 4 cm length and name it AB

Step (ii) Draw an angle equal to 105^o at point B. Draw and arc of radius 5 cm to draw side BC = 5cm.

Step (iii) At point C draw and angle equal to 80^o.

Step (iv) Draw and arc of radius 6.5 cm along CM and name it D to draw the side CD = 6.5 cm

Step (v) Join points D and C and complete the construction of the quadrilateral.

This is the construction of given quadrilateral.

Think Discuss and Write (2) We used some five measurements to draw quadrilaterals so far. Can there be different sets of five measurements (other than seen so far) to draw a quadrilateral?

The following problems may help you in answering the questions.

(i) Quadrilateral ABCD with AB = 5 cm, BC = 5.5 cm, CD = 4 cm, AD = 6 cm and ∠B = 80^o

Solution

First a rough sketch of given quadrilateral is drawn to estimate its construction.

Here to construct this quadrilateral the angle between side AD and AB and ∠D should be known.

Now, we know that, sum of adjacent angles of a quadrilateral is supplementary. This means sum of adjacent angles of a quadrilateral is equal to 180^o.

Thus, ∠ DAB + ∠ B = 180^o

Similarly, angle DAB and angle D are adjacent angles, consequently their sum will be equal to 180^o

Thus, ∠DAB + ∠D = 180^o

⇒ 100^o + ∠ D = 180^o

⇒ ∠D = 180^o – 100^o

⇒ ∠ D = 80^o

In similar way,

∠D + ∠DCB = 180^o

⇒ 80^o + ∠DCB = 180^o

⇒ ∠DCB = 180^o – 80^o

This means if opposite angles of the given quadrilateral are equal, then it is a parallelogram and hence opposite sides must be equal, which is not true in the given data.

Thus, without knowing the other two angles, it is not possible to draw the given quadrilateral.

Thus, construction of quadrilateral with given data is not possible.

Think Discuss and Write (2) (ii) Quadrilateral PQRS with PQ = 4.5 cm, ∠P = 70^o, ∠Q = 100^o, ∠R = 80^o, and ∠S = 110^o.

Solution

To draw the quadrilateral at least two adjacent sides and three angles must be known.

Since here only one side is given, thus construction of quadrilateral with given data is not possible.